fH3     ANALYTICAL     mUSmaSKtlOB     OP 
SLB3TRIC     BAIL7&Y    SPEED  -  TIME     ESATIONS 

By 

Lloyd  Hash  Robinson 

B.B.  (Union  College)  1911 
M.S.   (University  of  California)   1917 

THSSIS         '•-  •*•:««' 
Submitted  in  partial  satisfaction  of  the  requirements  for  the  degree  of 

DOCTOR     OF    PHILOSOPHY 

in  the 
GRADUATE     DIVISION 

of  the 

UNIVERSE?  Y    OP     CALIFORNIA 
June,  1919 


Approved  by  the 
Sub-CoEBiit  t  ee-in-Charge , 


Deposited  in  the 
University  Library, 


19  JQ 


Chairman. 


IOH Page    1 

I     FACTORS     HI     ELECTRIC     TRAIN    PROPULSION 

A  -  Energy,  Power  and  Tractive  Bffort     .......  3 

B  -  Cons>onont8  of  Tractive  Effort         ........  7 

Train  Resistance,  Curve  Reaistance, 
Braking  Effort  of  Friction  Brakes, 
Grade  Reaistance,  Accelerating  Effort. 

C  -  Train  Acceleration      .....*••••••••  11 

II     RAILY/AY    MOTOR     CHARACTERISTICS 13 

III     SPEED  -  TITJE     FORMULAE 

A  -  Train  Cycle     ..................  20 

B  -  Starting       ...................  21 

0  -  Normal  Terminal  Voltage  Applied  to  JTotora.   ...  24 
Balancing  Speed,  Acceleration. 

D  -  Coasting       ..•••••••*•••••••••  30 

E  -  Braking 36 

F  -  Sxwmary  of  Principal  Formulae     .........  39 

IV     EXAMPLES • 41 


•ft  I/-''" 


IOTHODUGTION 


?ho  iloai,-;;n  or  a  Modoi-n  olectric  rail./v  noceasarily 
an  acourato  predet  oral  ration  of  the  performance  of  the  completed  system 
whan  the  oonat motion  period  shall  hare  passed  and  the  traffic  begin* 
to  move*     Consequently,  the  speed  of  trains  predominate*  among  the  fac- 
tors to  be  treated*    The  speed  of  a  train,  lllce  that  of  any  moving  body, 
varies  as  a  function  of  time  in  accordance  with  the  fundamental  lairs  of 
mechanics.    From  these  principles,  it  is  possible  and  practicable  to 
jaBsdeterraine  the  speed  of  a  given  electrically  propelled  train  at  any 
instant  in  the  course  of  a  run  over  a  Known  or  asaroaed  track*    ?or  high 
speed  roads  with  frequent  stops*  such  aa  subway  systems,  the  converse 
problem  of  determining  the  time  required  to  attain  a  particular  speed 
may  be  of  greater  importance  and  of  correspondingly  more  frequent 
ooourr  ence • 

The  predeterminations  of  spoed-tlme  relations  necessarily 
depend  on  the  type  of  motive  equipment  and  therefore  must  bo  based 
upon  the  characteristics  of  the  motors  in  the  case  of  electrically 
propelled  trains*    The  methods,  in  current  use  for  carrying  out  these 
determinations  are  step-by-step,  graphical  processes*    The  most  used 


-1- 


. 

• 


.••; 


" 
-'. 


»«f  oc   ^Bfct;  e 
^J  MBO  orW 

. 

- 


method  is  that  proposed  by  Mr*  C.  0.  Liailloux  In  his  "notes  on  the 
Plotting  of  Speed-571n»  Curves"  In  the  Transact lona  of  the  /jaorican 
Institute  of  Sleotrioal  Sngineers,  Volume  XIX  (1902),  page  984. 

The  present  purpose  is  to  dorivo  formulae,  by  means  of 
the  spoed-tl*  relations  may  be  determined  directly  by  arithmetic  sub- 
stitutions and  operations  without  recourse  to  graphs  other  then  the 
characteristic  motor  curves*     Since  the  continuous  ouri-ent  aeries  motor 
Is  used  almost  universally  in  American  electric  railway  practice,  only 
the  formulae  applicable  to  this  type  of  motor  are  presented  although 
the  method  of  derivation  should  indicate  hoar  analogous  formulae,  suited 
to  other  types  of  motors,  may  bo  derived* 

As  a  foundation  for  the  subsequent  derivations,  a  segregation 
of  the  factors  involved  in  electric  propulsion  of  trains  is  essential* 


-  2  - 


c  ,    ;  vc  JkMoqp*!- 

10  •Illlt  Tri«ir>lT  «tft  r.l   ^MTWO  fOJ    - 

«  ••  ^RHI^'-  - 

IQ  tOMe:  •<»'  ,o*x£apt«ol  0tHaA  ot  ti  •io<tlWt  to 
ojt&  O'Slft   Awctettlftb  t>J  ^srt  a    . 


' 


I 

PAOTORS     DT     HLSflfBIC     T&XV    SROPULSlOsT 

A  -  gnSBPT.     POV.151    AITD    TRACT  171    BWOBg 

The  movement  of  a  oar  or  train  along  a  track  necessitates  an 
expenditure  of  energy.     In  the  general  case,  the  mechanical  energy, 
supplied  to  a  train,  simultaneously  servos  three  well  defined  purposes, 
and  consequently  must  be  treated  aa  the  sura  of  three  distinct  oangponents* 
These  ttxspasjfsfta  aret  first,  that  dissipated  tat  to  friction;  second, 
that  stored  or  liberated  as  potential  energy  due  to  change  in  the  eleva- 
tion of  the  train;  and  third,  that  stored  or  liberated  as  kinetic  mvegj 
due  to  change  in  the  speed  of  the  train* 

By  the  application  of  vihat  is  icoown  as  dynamic  braking,  that 
is,  operating  the  motors  as  generators  «hile  descending  grades  or  while 
Bstidng  stops,  mechanical  energy  may  be  extracted  froa  *  moving  train, 
converted  to  electric  energy  and  returned  to  the  electric  distribution 
system*    Ifiysloally,  the  extraction  of  energy  is  the  reverse  of  the 
process  of  supplying:  soargy.    Thus,  in  calculations,  energy  extracted 
must  be  treated  as  negative  energy  supplied*    Hence,  the  energy  supplied 
in  a  chosen  period  of  time  will  be  positive  or  negative  according  to 
operating  conditions* 

The  dissipation  of  enorcy  due  to  friction  continues  as  long 
aa  a  train  is  in  motion*    The  dissipated  energy  can  not  be  recovered  as 
useful  mechanical  energy  in  subsequent  operations  since  it  passes  off 
aa  heat  or  possibly  in  other,  less  easily  discernible  forms.     Its  nosja 
defines  it  as  lost*    The  dissipated  enercy  In  turn  may  conveniently 


• 
I 

• 

. 


- 


. 


. 


be  rosolvefi  into  throo  c exponents,  nts  aret  first, 

due  to  friction  ir.  trsv  •  •  l  tr.r<r  :*,  taol»i*»Uy  re- 

ferred to  as  tee-  to  trrln  roriot'nce:   noconfl,  tJtat  du<)  to  an  incrtcicnt 
of  friction  tetrodlnced  by  trriSh  crcrvrtnre,  usually  spfldosn  of  as  duo 
to  curve  resistance;  and  third,  thet  due  to  the  application  of  friction 
i,  Known  as  «tae~%t>  tanking  effc\  . . 

Potential  energy  it  stored  wbila  a  train  ia  ascending  a  grade* 
a  subsequent  descent,  potential  energy  is  liberated  and  either 
extracted,  dissipated  or  transfonasd  to  kinetic  energy*    kinetic  energy 
is  stored  v/nile  the  speed  of  a  train  is  increasing*    As  the  speed  later 
decreases,  kinetic  imrflj  is  liberated  sad  either  extracted,  dissipated 
or  trcnsformou  to  potential  energy,  depending  on  whether  the  decrease  in 
is  due  to  dynamic  braking,  friction  or  to  ascent  of  a  grade* 
Oxe  potential  energy  of  a  bodyAis  the  sans  ^>en  it  is  at  tto 


,  ,  —  -  ,    •  »>rt-<rVT^' 

elevation*  Also  the  kinotio  energy  of  a  body  at^  steadstill  is  zero. 
TlMirnifri  n  all  the  energy*  that  in  supplied  to  a  train  during  its  run 
between  two  stations  ,7hioh  are  at  the  sons  elevation,  is  ultimately 
dissipated  before  the  train  stops  unless  some  of  it  is  recovered  by  means 
of  dynamic  braiding* 

Although  the  foregoing  discrimination  of  the  forms |  in  which  . 

—.  »T 

energy  is  expended,  is  an  essential  step  in  tho  analysis  of  train  pro- 

f> 

pulsion,  the  term,  speed-time  relation,  accur&toly  lilies  instantaneous 

values  of  speed  and  time*    Consequently  the  determination  of  these  re- 
lations is  iiiniouiatoly  concerned  with  insJtentTjqoous  rates  of  energy 
expand iture,  conawqption  and  absorption^    'Jhat  is  to  say,  it  depends 
H>on  the  input  and  distribution  of  mechanical  power,  for  power  is  the 
instantaneous  rate  of  energy  expenditure,  conversion,  consumption  or 

-4  - 


IS   t< 

••  . 


. 


absorption.  Tho  mechanical  energy  input  ha*  bean  divided  into  three 
main  OQBPponenta.  Similarly,  the  mechanical  power  input  is  the  sum  of 
the  corresponding  instentanooua  ratea  of  energy  dissipation  and 
storage* 

The  measure  of  mechanical  power  input  to  a  moving  body  is 
tho  algo  ornie  produot  of  the  applied  force  and  the  instantaneous  speed 
at  which  the  point  of  application  of  the  force  io  moving*  In  an  elec- 
trically propelled  train,  the  mechanical  power  input  is  transferred 
from  the  motors  throi&i  sears  to  tho  axles  at  the  surface  of  the  latter* 
Hence  the  measure  of  the  mechanical  poerar  input  to  an  axle  is  tho  pro- 
duot of  the  tangential  force  at  the  surface  of  the  axle  and  the  peri- 
pheral speed  of  that  surface*  However,  in  railway  calculations,  it  is 
convenient  to  use  the  speed  of  the  train  aa  a  basis  of  reference  so  far 
aa  possible* 

flu  peripheral  speed  of  a  point  on  the  tire  of  &  oar  wheel  is 
flat  MM  as  the  speed  of  the  oar  provided  the  wheel  does  not  slip*  HJhe 
peripheral  speed  of  a  point  on  the  tire  of  a  wheel  is  to  tho  speed  of 
a  point  on  the  surface  of  its  axle  in  tho  ratio  of  the  diameters  of  the 
wheel  and  axle*  Therefore  the  ratio  of  the  train  speed  to  the  peripheral 
speed  of  tho  cyolindrioal  surface  of  the  axle  is  equal  to  the  ratio  of 
the  diameters  of  wheel  and  axle* 

By  the  principle  of  moments,  a  tangential  force  at  the  surface 
of  an  axle  may  be  replaced  by  an  equivalent  force  applied  at  t  ho  surface 
of  a  concentric  oyclindor  of  greater  radius*  Zhe  tangential  force,  which 
must  be  applied  at  a  radius  equal  to  the  wheel  radius,  in  order  to  pro- 
duce the  Maw  moment  aa  the  actual  tangential  force  applied  at  the  surface 


-  5  - 


9*  Of 


of  tho  axle,  must  be  to  the  latter  In  the  Inverse  ratio  of  tfw  diameters 
of  the  wheel  cad  axle*    This  equivalent  force,  hypothetically  noting  on 
the  axle  at  a  radius  equal  to  the  serai-dieTnotQr  of  e.  driving  wheel,  ia 
•ailed  the  tractive  effort* 

She  above  relations  can  be  expressed  concisely  in  algebraic 
form:  thus 

(  Mechanical  power  }  {  Tangential  force  applied  )  (  peripheral  speed  ) 
(  input  to  train  |  •  (  at  aurfaoe  of  a  driving  )  X  (  of  cyclindrical  j 
(  per  motor  (  axle  }  (  surface  of  axle  ) 

(  Tangent  ir,l  force  applied  )       (  peripheral  speed  ) 

(  at  surface  of  a  driving  )       (  of  axle  surface     ) 

(  axle  X  axle  diameter  [  *  I  I  «he«l  diameter  ) 

(  i   vjhoel  diameter  )       (  i    axle  diameter  ) 

•    Motor  tractive  effort  X  train  speed  • 

Tli&  stnndard  /onerioan  unit  of  measure  of  train  speed  is  miles 
per  hour,  and  that  of  tractive  effort  Is  pounds,  avoirdupois*    25io  total 
tractive  effort  applied  to  a  train  is  tfce  sm  of  the  tractive  efforts 
applied  at  the  several  driving  axles*    Hovever,  for  purposes  of  com- 
parison and  for  simplicity  in  computations,  the  tractive  effort  is 
usually  referred  to  the  welgit  of  the  train*    That  is.  tho  tractive 
effort  is  spoken  of  as  so  many  pounds  per  ton  of  gross  train  weigit*  The 
phrase ,  pounds  pex*  ton,  unfortunately  contains  a  latent  ambiguity  because 
tho  number  of  pounds  in  a  ton  Is  preranably  constant,  but  there  should  be 
no  confusion  \rhen  the  expression  is  confined  to  railroad  parlance* 

Because  of  tho  proportionality  of  mechanical  power  aad  tractive 
effort,  the  applied  tractive  effort  is  obviously  divisible  into  several 
components  corresponding  to  tho  rates  of  energy  dissipation  ®nd  storage* 


• 


In  geoex1?-!.  the  trrctlvo  effort  applied  to  a  train  jaay  be 

»  j  -~~V 


rosolvoii  Into  the  foil  rains  consonants  t 

1  -  Oorroaponiline  to  the  rate  of  enercy  diasipatlon, 

(a)  -  Train  resistance, 

(b)  -  Curve  resistance, 

(o)  -  Braiding  effort  of  friction  brakes; 

2  -  Corresponding  to  the  rate  of  storage  of  potential  energy, 

(a)  -  Srado  resistance! 

3  -  Corresponding  to  the  rate  of  storage  of  Kinetic  energy, 

(a)  -  Accelerating  effort* 

•:*  « 

rescB 


V«nen  a  tr&iu  ic  moving  on  Icvol  tangent  trf.c';,  thoro  10  energy 
dlaslpatc'.l  In  rail  raid  journal  friction,  eir  resistance,  etc*    She  son 
of  the  forces,  corresponding  to  tho  rates  of  energy  dissipation  due  to 
these  causes,  is  c?  lloa  the  train  resistance*    2h«  •agaltado  of  the 
train  resistance  at  any  inattuit  dopends  tepon  the  train  weight,  speed, 
cross-sectional  area,  and  the  mc&er  of  oars*    The  relations  of  these 
factors  have  been  determined  by  oxtencivs  eaqportooiite  with  diff  ercmt 
Classen  of  oqidptnent  in  operations  under  wide  ranges  of  conditions*  Prom 
ftM  oaqperlmental  data,  empirical  fonanlae  for  trr-in  resistance  hiive 
boon  deduced*    One  of  these,  flfeidx  10  quite  comonly  uaoii,  is  tfu:t 
developed  by  Mr*  A*  H*  Armstrong, 


In  whidh 

R  »  Train  resistance  In  poonfla  per  ton  of  gross  train  wel&it, 

T  *  (Jroan  wel^it  of  train  la  tons  (3000  lbs«), 

7  »  3peod  of  train  in  miles  per  hour, 

X  »  Projeotec!  croas-sectional  area  of  train  In  sqwve  feet, 

H  «  ITamber  of  cars  In  train, 

>  3.3 

VaT 

-  7  - 


V 


'      •  l' 


. 


- 


Quire 


torixoat 


Vhen  a  trr.in  is  proceeding  along  a  horiaontal  curve  in  tho 
trade,  there  is  introduced  an  additional  resistance  due  to  the  side 

jsurs  of  tho  ufaecl  flanges  on  the  rails,  tho  unequal  distribution 
of  weight  of  t&c  trela  on  the  two  rails,  etc*  This  additional  reals- 
is  called  curve  resistance* 


DM  curvs  real seance  varies  from  0  =  0*5  D  to  0  •  1*5  D 

.,  .. 
pounds  per  ton  of  gross  train  weight,  depending  upon  the  condition 

of  tho  tracfc  and  tihcela  and  vg>on  ttu»  degree  of  a^eroleration  of  the 
outside  rail*  D  is  the  degree  of  tho  curve j  that  ia,  ,ho  ntcd>or  of 
degrees  of  central  angle  subtended  by  a  one  hundred-foot  chord  of  the 
circular  arc  described  by  the  center  line  of  the  track*  It  la  clear 
that,  for  average  conditions,  0  may  be  talon  as  mmsrioally  equal  to 
the  degree  of  the  curve. 
Braking  Hffort  of  Friction 


friction  brakes  are  applied  to  the  tfeeeola  of  a  moving 
train,  energy  is  dissipated  in  accordance  with  the  lava  of  aliuing 
friction  of  natal  on  metal  tdder  pressure-  However,  since  the  pressure 
of  the  braksa  shoes  is  subject  to  the  iraaadi&to  control  of  the  operator, 
tho  magnitude  of  tho  braking  effort  does  not  bear  any  fixed  relation 
to  the  train  speed*  In  applications  of  tho  braioss  on  long  domgrades, 
the  braking  effort  is  iwpt  practically  oonstoitj  and,  when  service 

stops  are  being  made,  tho  braking  period  is  of  such  short  duration 
that  no  serious  error  in  con^uations  is  introduced  by  treating  the 
braking  effort  aa  constant  during  this  period*  Considerations  of  tho 
aafety  of  tho  equipment  and  of  the  comfort  of  passengers  dictate  the 


-  a  - 


allowable  braking  effort*  In  subueouent  formulae,  the  bralcing  effort 

of  friction  brakes  is  represented  by  B  and,  liice  the  other  component* 

of  tractive  effort,  it  is  measured  in  pounds  por  ton  of  gross  train 

weight. 

Grado  Hesistanoe 

In  mounting  a  grade,  a  train  absorbs  energy  which  is  return- 
able when  the  train  later  descends  to  a  lower  elevation*  Ihe  component 
of  tractive  effort,  corresponding  to  the  rate  of  storage,  or  liberation, 
of  potential  energy,  it 

a  -  2000  sin  j*  , 

where  2000  is  the  number  of  pounds  in  a  ton,  and  $  is  tho  vertical  angle 
measured  upward  from  the  horizontal  to  tho  grade  line  of  tho  track* 

Since  the  per  cent  grade,  100  tan  /J,  can  readily  be  obtained 
from  the  surveyors*  data,  it  is  convenient  to  express  tho  grade  resis- 
tance as 


0  -  2000  sin  (  tsfi1  (  ?a*  ««*  grade  1  )  4 
(     (     100      )  ) 

Tor  light  grades,  such  as  are  met  in  steam  railroad  practice,  there  is 
no  appreciable  error  introduced  by  assuming  that 

•in  0  *  tan  0 
and  using  the  approximation, 

a  *  0'  -  2000  (  ^'  *g*  *****  )  -  20  X  per  cent  grade. 
But  ranch  heavier  grades  are  common  in  urban  and  suburban  electric  sys- 
tens  and  serious  error  may  be  introduced  in  calculations  by  applying 
the  approximation* 


-  9  - 


• 


While  train  roaiatanoo,  OUFTO  resistance  and  the  explication 
of  friction  braicoB  all  tend  to  retard  tho  motion  of  a  train,  It  la  evi- 
dent ISiat  grade  resistance  will  tend  to  retard  or  to  aocolerato  depend- 
ing on  whothor  the  train  la  proceeding  uphill  or  down.  OJhia  la  taicon 
Into  aooount  In  the  formulae  by  assigning  to  the  value  of  G  the  positive 
or  negative  algoUraio  algtx  ae  diotatod  by  the  above  oonaiderations. 
Chat  Is  to  say,  the  vine  of  G  la  positive  or  negative  acoordlng  aa 
that  of  ain  fl  la  poaitivo  or  negative* 
Accelerating  Effort 

If  the  magnitude  of  the  propelling  force  applied  to  &  body  la 
greater  than  that,  undor  tho  action  of  which  the  exlatlng  apoed  of  the 
body  will  bo  jnalntainod,  the  speed  of  the  body  will  increase;  and,  con- 
versely, the  apoed  will  flecroaae  if  the  propelling  force  la  Insufficient 
to  maintain  the  ocfruit  apeed  of  the  body*  Consequently,  If  the  tractive 
effort,  applied  to  the  driving  wheola  of  a  train,  la  more  than  sufficient 
to  ovorooiae  the  train  reaiatance,  curve  roa  is  fence,  braking  effort  and 
grade  raaiatance,  tlio  balance  will  operate  to  accelerate  the  apoed  of 
tho  train*  2hia  conpouont  of  the  tractive  effort  nay  be  termed  the 
r ocol orating  effort* 

The  accelerating  effort,  neoeaaary  to  produce  an  acceleration 
of  A  miles  per  hour  per  second,  la  usually  taJoan  aa  100  A  pounda  per 
ton  of  groaa  train  weight*  The  value  of  the  accelerating  effort  la 
poaitive  or  negative  depending  on  whether  that  of  A  la  poaltive  or  nega- 
tive; thE-t  is,  according  as  the  speed  of  the  train  is  increasing  or  de- 
er easing. 


-  10  - 


".    >Y: 


. 
. 
' 

.-"(I 

i      ; 

• 

•      . 


of  Oom^oaBnts 

The  algebraic  sm  of  the  above  five  components  is  the  tractive 
effort  applied  to  the  driving  axles  through  the  functioning  of  the  motors 
and  gears*  That  is  to  say, 

P  «  XOO  A+B+C+G+R    , 
in  which        P 


• 


B 


I 


Tractive  effort  applied  at  driving  axles, 
Acceleration  of  train  speed  in  miles  per  hour  per  second, 
Braking  effort  of  friction  brakes. 
Curve  resistance, 
Grade  resistance, 
resistance* 


The  unit  of  neasure  of  all,  except  the  acceleration  A,  is  pounds  (avoir 
dupois)  per  ton  (2000  Ibs.)  of  gross  train  weight* 


0  -  TRADT 


Transposing  7  and  A  in  the  above  tractive  effort  equation, 

A  -  0*01  (F-3-C-a-n)  • 

This  relation  expresses  the  fact  that  the  acceleration  is  determined  by 
the  values  of  the  applied  tractive  effort,  braiding  effort,  ant  curve, 
grade  and  irain  resistance* 

^f  o_-  \M**  .**• 

Die  acceleration  is  the  instogfrtaeom  W*+  rate  of  Change  of 

A^-ttv*!  'fcifc*' 


the  train  speedy  tiiat  is, 

dV 

JL     «t     n 

dt 


Henco  -          «.  o.Ol  (P-S-C-a-a)  • 

dt 

Substituting  for  B  its  equivalent,  given  on  page  7  above, 

,1) 


-  11  - 


jjj  frfrlt . 

, 


J 


This  relation  (1)  la  the  fRr.damentr.1  fltff erratic!  •cjnation 
of  train  spa©*.     Its  solxttionB  reader  formolae  for  the  direct  |fl?e- 
aotermlurtion  of  tho  ej>0cv*-«t.l"w  relations  for  kBOm  or  agmmed  «w- 
vloe  conditions*    As  ha0  l»oen  provtously  stated,  3,  C  and  a  are  inde- 
pendent of  tho  train  speed  Y«    However,  the  applied  tractive  effort 
F  »y  vary  as  A  function  of  the  train  speed  as  will  "be  seen  from  a 
cons iflorat ion  of  tha  <£r>.ractoriatic  curves  of  railway  motor •  and  the 
natal  methods  of  operatix£  the** 

'       »-.    W.V.  ••-• 


-  12  - 


XI 

2AXL1AT  MOTOR  CKARACTJ2RI37IC3 

Che  ear  lea  motor  derives  its  name  from  the  fact  that  its 
field  circuit  is  connected  in  soriaa  with  the  armature  circuit  aa  is 
shown  in  Pig*  1*  It  la  clear  that  tSio  motor  current,  armature  cur- 
rant and  field  current  are  identical* 

The  performance  of  a  motor  under  operating  conditions  is 
moat  conveniently  expressed  in  vfeat  are  teaown  aa  characteristic 
curvoa.  Tioae  ourvoa  for  railway  motora  are  determined  from  their 
performance  in  teata  which  are  usually  made  at  the  factory*  Fig*  E 
shows  tbe  oharaot eristic  curves  of  a  continuous  current  aeries  rail- 
way motor*  For  normal  voltage  applied  to  the  motor  teminfus,  these 
curvoa  display  the  relations  between  the  motor  current  and  each  of 
the  following  fact  or  at 

1*  Speed  of  oar  or  train  in  milea  per  hour* 

2*  Tractive  effort  of  motor  in  pdnqfls, 

3*  Efficiency  of  motor  and  ita  gears  in  per  cent, 

4*  Power  output  of  motor  with  ita  goora  in  kilowatt  a* 

It  ia  clear  that  the  relations,  expreaaed  by  the  curves,  are 
all  affected  more  or  leaa  by  the  diameter  of  the  driving  wheels  and  by 
the  reduction  ratio  of  the  gears  between  motor  and  occlo*  Also  tho  re- 
lation of  apeed  to  current  la  largely  dependent  upon  tho  motor  tormina! 
voltage*  For  those  reaaona,  the  gear  ratio,  nheel  diameter  and  terminal 
voltr-£o,  for  which  the  ourvoa  apply,  aro  stated  on  the  curve  aheet* 

Tho  ap««d-current  curve  shows  that  tho  current  decreases  aa 
the  apeed  increases.  2hla  is  due  to  the  direct  proportion  connecting 


-  13  - 


fin 


•  dtoa  M»  i  .«fe9$»6  -{X«f 

-TO  «tf  ac  feNto^i  M*  ,v;Xi-;'«  Mrrwb  M*  *JA 
••  »«4«s0-  os*  J4i»tm>  wft  (NcO  WTJOA  ?T«JO»  va»*i«ro-Jkie^» 

lit-   MV  «f   trft     :?1    Oi/T 


To  controller 
and    supply 


FIG.   I 

SCHEMATIC    WIRING     DIAGRAM 

OF 
CONTINUOUS       CURRENT    SERIES    MOTOR 


CHARACTERISTIC   CURVES  OF  A  '   .',   : 
CONTINUOUS   CURRENT    SERIES    RAILWAY 


the  apeed  of  rotation  of  th«>  armture  with  the  tartueei!  counter-electro- 
motive force  of  a  motor.  Wxst  ia  to  aay,  an  inoreaae  of  the  afwature 
apeefl  produoea  an  increase  of  the  electromotive  force  induced  in  tho 
armature  and  this  oppoaen  the  impreaeed  electromotive  force,  tht»  re- 
ducing tho  current  in  the  motor  circuit*  Tho  curvea  ahow  alao  that 
the  tractive  effort  decreaaea  aa  tho  current  decreaaea*  ffl»6  nat  reault 
is  that  the  tractive  effort  input  to  the  driving  axlea  deoreaaea  a*  the 
•peed  of  the  trr.in  inoroaaea*  In  other  wor*s,  tho  tractive  effort  bear  a 
a  fixed  relation  to  the  apeod  aa  long  aa  constant  voltage  la  applied  to 
the  terminala  of  the  motor*  Thla  relation  mat  bo  determined  ao  that  an 
expreaaion  for  tractive  effort  in  terms  of  apoed  can  be  aubatituted  for 
F  in  the  differential  equation  (1)  before  the  latter  can  be  aolvod. 

Although  tho  apeed-ourrent  and  tractive  effort-current  curvet 
indicate  the  existence  of  perfectly  definite,  continuous  relations  bo- 
ttreen  speed  and  tractive  effort,  it  ia  fcq?oaaible  to  expreas  these  re- 
lationa  in  a  formal;:  rationally  derived  from  fundamental  principles. 
The  alternative  ia  to  uae  an  approximation  that  ia  aufficlently  exact 
for  engineering  purposes* 

The  tractive  effort-current  and  power  output-current  curvea 
in  Pig.  2  —  and  these  ourvoa  are  typical  in  this  reapect  although  they 
apply  to  a  particular  motor  —  may  be  oloaely  approximated  by  the 
atraigat  llnea  AA  and  BB  over  the  operating  range  of  tho  motor*  It  haa 
already  been  ahoron  that  the  power  input  to  the  train,  which  la  equal  to 
the  power  outputs  of  all  tho  motors  combined,  ia  the  algebraic  product 
of  the  tractive  effort  and  train  apeed.  3y  making  uae  of  those  relations 
and  the  above  straight  lino  approximationa,  an  algebraic  expression  of 
the  relation  between  tractive  effort  and  apeed  aay  be  obtained* 


-  14  - 


. 


Alt*  ftttEl 


>-.j2 


• 


equation  of  the  line  AA  in  Pig*  2  la 

P»  -  h}  *  hjl  (2) 

in  vhitih 

P*  B  Power  output  per  motor  in  kilowatts, 

I    •  Current  per  jnotor  la  as$}3rost 

h*  and  b*  are  constants  determined  by  the  co-ordLrv.tou  of  any 
ttro  poJnta  on  tiio  lino  AA* 

To  roduoo  the  payer  to  kilowatts  per  ton  of  groae  train  weiflfrt,  «ib- 

atUute 

-1-p  .  pt  (3) 


In  v^iioh 


P  »  mecfhanloal  power  input  to  driring  axles  In  Kilowatts 

per  ton, 

T  -  Oroaa  weigit  of  train  in  tons, 
H  •  Vonb*r  of  rootore  in  the  tr-1  in. 

P  -  h|  +  h|l  (4) 


or  P--|~th|  +h^I  )  (8) 

Since  ?  is  e2pre«aed  in  pounds  per  ton  and  V  in  mile*  per 
hour,  and  ainoo  it  Is  doaired  to  ezpreea  tho  power  in  term  of  ?  and  Vt 
the  potroiv  *  kUowatta  per  ton.mnat  be  retnoea  to  mile  pounds  per  hour 
per  ton*    One  idlowstt  la  oqulTe,lont  to 


503  »    980  *  36W-   mile  pounda  per  hour.  (6) 
0*746  X  5280 

Thus  the  poorer  input  to  the  driving  axles  la 

y  V  "  503  P   mile  pounds  per  hour  per  ton*  (7) 

:  J  (8) 

V-.B03Mh{  (9) 


h.* 


-  15  - 


1 

I 

1 


. 
' 

' 

. 

t.  .;  •ftf   a?3*K;Sri 
1  «aar.T.     oil:  . 

-   ..iw     •jf.«v:- 


.  "O*  1»^[  «rr 


The  equation  of  the  atrai^it  line  BB  in  ?ig«  2  ia 

P1  -  h3  +  h4I  (10) 

in  which 

P*  •  Tract  ire  effort  output  per  motor  in  pounda, 
I  =  Our  rent  per  motor  in  ao^eroa, 

h,  and  h4  are  constants  determined  by  the  eo-ordinatea  of 
any  two  points  on  the  line  3B« 

To  reduce  the  tractive  effort  to  pounds  per  ton  of  gross  train  woi&xt, 
substitute  -2-  F  -  7*  (11) 

rfMft  >r         M 

In  which 

P  •  Tract  ire  effort  input  to  driTiug  axles  in  pounda  per  ton, 
TV}          I  »  Gross  weigpit  of  train  in  tons, 
H  •  Umber  of  motors  in  train. 

Then 


or 


JL  p  -  hg  +  h4l 


T     (  h4T  -  503  hj) 


503  M 

T      "(  V-  503h|/h4) 


«  P  -  It  h, 

-    -  tui 


Combining  equations  (9)  and  (14), 

t  T  Y  -  503  II  hj  T  F  -  II  hg 

503  Mh!  Mh.  * 

2  •? 

Then  *  P  V  a4  -  603  M  hjhj^    -    SOSTPhg-SOSH  hjig     ,  (16) 

*  (  H4V  -  503  hg  )  P    -    503  M  (  hjh4  -  n|hg)       ,  (17) 

503  M  (  ^A  "  h?*3V.  ,  (18) 


-  16  - 


• 


' 


'' 


(   rf  «oa  -  T,.- 


•« 

Letting  hj  -  ™   (  h|  -  V^  )  («» 

t         4 

h2 
and  hg  -  503  —     .  (21) 


F  -  --     .  (»> 


aquation  (22)  1*  the  approximate  formula  for  tractive  effort 
F,  in  pounds  per  ton  of  gross  train  weiffct,  in  terras  of  the  train  speed  V» 
in  miles  per  hour,  vfrum  rated  voltage  is  applied  to  the  terminals  of  the 
motors* 

?or  reasons,  that  will  appear  presently,  the  tormlnal  voltage 
of  tfro  motors  is  roauoed  below  the  rated  voltage  during  certain  periods 
in  the  operation  of  a  train*  In  fact,  there  are  three  conditions  of 
terminal  voltage  to  be  oonaideretl  ,  namely  t 

1*  Operation  at  rated  voltage) 

2*  Operation  with  power  shot  off,  that  is, 
with  aero  voltage  applied; 


3*  Operation  with  applied  voltage 
rated  voltage  and  not  aero* 


AM  has  bean  pointed  out,  equation  (13)  applies  in  the  first 
of  those  throe  oases* 

While  a  train  is  moving  with  tbo  por/or  shut  off,  13io  terminal 
»,  applied  to  the  motors,  is  zero;  henoo  the  motor  current  is  sero* 
output  of  tho  raotoai  however,  is  slightly  negative*  That  is,  a  small 
amount  of  energy  is  extracted  from  the  train  and  dissipated  due  to  fric- 
tion in  the  motors  and  gears*  However  the  rate  of  extraction  of  thia 

is  so  snail  that,  for  practical  purposes,  it  is  neglected  so  that 

F  «  0  (23) 


-17  - 


• 


, 


. 


Bceept  rthilo  starti^,  trains  ere  seldom  operated  for  any 
appreciable  length  of  time  with  any  terminal  voltage  loss  than  normal 
applied  to  the  motors*  In  spools!  oases,  a  oonatant  partial  voltage 
nay  be  applied*  But  such  oases  are  rare  and,  vfaon  they  most  be  treated, 
an  erprooalon  for  ?  in  the  form  of  ecuatlon  (22)  can  readily  be  derived 
by  making  suitable  modifications  of  the  motor  characteristic  onrves  to 
accord  with  the  partial  volt^o* 

Before  a  train  starts,  tho  speed  of  the  motors  is  sbro,  so 
the  electromotive  force,  induced  in  the  armature,  la  «ero.  Hence,  if 
voltage  is  applied  to  the  motor  terminals  with  tho  train  at  stand- 
still, the  current  is  liTnitce  only  by  the  resistance  of  the  field  sad 
armature  circuits  combine!!*  This  resistance  is  neoosearily  so  small 
in  railway  motors  that,  if  the  rated  voltage  were  applied  to  the  motor 
terminals  while  the  train  was  at  standstill,  the  motors  would  either 
be  d-juagod  seriously  or  develop  an  excessive  torcfue  causing  the  wheels 
to  slip  or  the  train  to  start  trith  a  severe  jerlc*  (Therefore  the  motor 
terminal  voltage  is  varied  so  as  to  keep  the  currant  within  safe  limits 
during  the  starting  period* 

Ihe  voltage,  applied  to  a  train+ls  usually  constant  so.  in 
controlling  continuous  current  series  railway  motors,  the  adjustment 
of  notor  terminal  voltage  is  procured  liy  connect*^  external  resistance 
in  series  -wittx  the  raotora.  From  several  points  of  vlowt  the  ideal  con- 
trol would  maintain  tho  motor  current  constant  at  its  muxlsRxa  permissible 
value  throughout  the  starting  period*  For  practical  reasons,  however, 
Hie  ecatom  is  to  vary  the  extcra^l  resistance  In  a  fev  steps,  keeping 
tiae  current  within  certain  well  defined  limits  v/hilo  starting*  Although. 


•      . 


MkitM 


, 


it  Is  not  maintained  by  thin  raothod,  it  is  practicable 
to  treat  the  our  rent  as  though  it  were  constant  at  an  equivalent  aver- 
age  value.  It  haa  been  sh.ovm  that  the  motor  tractive  effort  dopende 
«pon  the  notor  current  and  is  independent  of  tho  motor  terminal  vol- 
tcge  except  in  ao  far  as  tho  letter  affbcte  tho  magnitude  of  the 
current*  Hence,  ?m  equivalent  constant  starting  current  produces  a 
corresponding  constant  tractive  effort*  In  other  words,  tho  tractive 
effort,  ?  in  equation  (1),  is  treated  as  constant  from  the  instant 
nhen  tho  train,  beting  to  move  until  the  train  spoei  attains  the  value 
indicated  in  the  motor  characteristic  curves  as  corresponding  to  the 
averse  stsurting  current* 

Tilth  tho  permissible  average  startiag  current  fixed,  the 
corresponding  tractive  effort  is  determined  from  the  characteristic 
curves.  This  constant  tractive  effort,  expressed  in  pounds  per  ton, 
i  tho  value  of  P  in  equation  (1)  during  the  starting  period. 


-  19  - 


. 


' 


Ill 

MB  -TIM    POBMUUK 


The  ti'.-.lui.  of  a  railway  syaton  operate  in  Mtoat  are  called 
trips  between  b&iviL-uile,  vjhich  may  be  at  the  ondo  of  s  lino  of  tra0fc 
or  Intoroediato  as  at  the  anfia  of  lUvloioua.     In  the  course  of  a  trip, 
a  train  usually  si&lcoa  aovaral  stops*    The  period  of  time,  that  olnpaM 
botwoon  r/^ion  a  train  lonvtMi  ona  atopplag  place  and  vhon  It  Icavoa  the 
nooct  t  is  oalleo.  a  train  cyolo* 

The  rirst  Gvoiib  la  a  train  cycle  la  the  at;  rt ,  igiiich  is  f ol- 
lotrod  by  •xiocoasiTo  perioils  of  opera*  ior.  oach  imolTiaer  nor*  or  less 
different  com"!  it  ions,  •  ud  finally  cones  tha  stop*    /ll  tha  normal 
conditions  of  operation,  thnt  r.ro  not  in  a  train  cycle,  way  be  olaeei- 
fiod  tciilor  a  few  typioal  phaoat  tvhloh  ore  ooaed  largely  in  accordance 
with  the  effects  tha*  they  prodtuso  tgjon  the  train  apcot!*    They  aro» 

X.  start  log, 

2.  Accoloratine, 

S*  I5tc'»uing  at  constant  spood, 

4.  Coasting, 

5«  Braking, 

6*  Stop* 

BMh  of  these  phases  of  operation  entails  a  unique  interpret  at  ion  of 
certain  terms,  particularly  the  input  tractive  effort  F,  in  the  funda- 
mental differential  equation  (!)•    And  upon  these  interpretations  rest 
the  solutions  of  this  equation  and  thereby  the  formulae  for  the  speed- 
relations. 


-  20  - 


For  obvious  reasons,  the  braiooa  are  not  ordinarily  applied 
while  tho  motora  are  functioning  and  consequently,  during  tho  greater 
part  of  any  train  cycle, 

I  -  0  .  <f* 

Also  curves  and  grades  are  of  incidental  occurrence  ao  moat  train 
cycles  inclttf  o  periods  when 

0-0,    G  =  0   or   C  »  G  "  0  • 
.'.hen  B,  C  or  G  la  not  aero,  it  nay  be  constant |  but,  if  it  is  not 
constant,  16  ie  usually  nocossary  and  sufficiently  accurate  for  pruo- 
tiot-1  purposes  to  treat  it  aa  constant  by  taking  average  values  over 
definite  intervals.  Xhe  terma,  B,  0  and  G,  will  be  carried  aa  constanta 

in  the  present  dor ivat ions  in  ordor  that  the  resulting  formulae  nay  be 

. 
univer aally  applicable. 

Having  determined  tho  tractive  effort  for  the  three  poaaible 
conditions  of  applied  motor  voltage,  and  having  divided  the  train  cycle 
into  its  component  phases  of  operation,  it  remains  to  eolva  the  above 
equation  (1)  for  enoh  of  those  several  phasea* 


It  has  been  pointed  out  above  that,  during  the  starting 
period,  the  voltage  applied  to  the  motor  terminals  is  varied  ao  aa  to 
maintain  practically  constant  tractive  effort.  This  permits  tho 
solution  of  equation  (1)  on  the  assumption  that  F  is  constant. 


-21- 


f  .' 


to 


ta- 


. 


: 


•V   I 


In  order  to  facilitate  manipulation,  group  the  constants 
and,  for  the  starting  period,  let 

/O,  =    O.OOO3    , 

„         0.00002  X  /,  .   N-l  ) 
7,  r  V      /o   J 

edition  (1)  becou»8 


dt 
dt   = 

1,dt    = 


. 


ctV 


Slnoe  --is  constant. 


Integration  of  this  equation  renders 


I 


(26) 

(27) 
(23) 
(B9) 
(30) 


(33) 


(34) 


where  C'  is  a  constant  of  integration.  Dividing  both  members  of 
(34)  by  tf  , 


_c: 

Transforain;  to  comon  loearithma  in  order  to  facilitate  confutation, 

/     .                     ^.00             7^^,        1 3d-'  ''    -'  w  -"*     I     y._£o_  /3S) 

— /   .  ^=%  '  °S'0     I       /  .     .  .,     3-5        _  _  _,  .  ,  \  *  \.-~>f3) 


V. 


-  2S  - 


A= 


(St) 

(ee) 


*v: 

•• 


^  ^x  \ 

UN. 


1.1 
\0 


(»> 


It  the  train  speefi  la  7.  at  the  instant  t  ;  tho  tine  t,  In 


starting  period,  at  tfilch  the  apeefl  will  reach  saay  other  vslue  V, 
it  given  "by 


2.30      ^     .  ^   y<M,t  +/3?--t-/3,+z7,  V.        ^          ^    < 
*         "z  ' 


tl-.i.l:    IB, 


^aaraiLl  I 

^^/«/-/0,-^^  J  J 


or 

f  = 


log,0  (V4<x,<t-r#*  +/3,+ay,  V)  - 


w 


t  and  t  are  aspraaaetl  In  secoMa  and  Y  and  V  in  miles  per  hour 
Combining  equations  (36)  and  (36),  and  letting 


C, 


- 


It  la  Men  that 


or 


equation  (39)  reduces  to 


«  0  ,  t 


If  a  train  is  started  from  atrndatlll  and  ti  o  la  aaanred 
from    1h«  Inatant  of  atarting,  ao  tfc&t  YQ  "  0  and  tQ  "  0  |  •qwttion  (38) 
that  tiie  ntniber  of  seconds,  reqtdrod  for  the  train  to  attain  the 

V  miles  per  hot*,  la 

.  23  - 


I»JK 


9S' 


v  t 


.     • 


4  - 


2.30 


j 

*  "    ~ 


, -A V- 


,      \ 


(46) 


mT.n«Aiia    ivotiTn, 


At  the  conclusion  of  the  nbove  starting  poriod,  \*.an  foil 
rated  voltaffe  la  appliofl  to  the  motor  terminals,  the  motor  prooooda 
to  function  In  accordance  with  its  dharooteristlo  curves.     Ordinarily, 
there  will  T>e  a  poriod  of  acceleration  between  th«  «r»d  of  the  start- 
Ing  period  and  the  time  when  fall  speed  is  attained.    Also,  in  subse- 
q-uent  periods  of  a  train  cycle,  there  may  be  acceleration,  positive 
or  negative,  as  grades,  curves,  etc*  arc  encountered*    As  long  as 
rated  terminal  voltage  is  applied  to  the  motors,  the  relation  of 
tractive  effort  and  speed  is  approximately 


Substituting  thla  In  equation  (1)  glvos 

dV 
-dT  =  °'° 


-0.03V- 


/O 


o.o/ 


V-h, 


(22) 


-  24  - 


-j^V' 


i  \  \  v  ^.iwv  c\ 

&- 


vs*;}- 


Let 


dt 


Let 


c£  =  o.o/  F 


o.ooax  ,      /y-/ 

7-  ('  +      /O 


A 


O. 


Q.QI 
<G 


o.oah*  - 


1  - 


In  railway  parlnnce,  a  train  operating  at  oozutaat 
fated  voltage  applied  to  the  motor  terminals  la  apoloen  of  &• 
naming  at  balnnolng  speed*  The  term  arises  from  the  fact  that 

/ 

iJhQn  a  train  i«  operating  at  oonatant  speed,  the  power  li^ut  is  Jtiat 
balancoi:  by  the  power  dissipated  plus  the  rate  of  storage  of  potential 
energy*     In  other  words,  thoro  is  no  kinetic  energy  being  stored  in, 
or  extracted  from,  the  tmin* 

Since  the  speed,  V  in  the  foregoing  equations*  becomes  con- 
stant at  the  balancing  spoed,  the  latter  can  be  designated  as  a  parti- 
cular value  of  7,  say  7«  • 

Also,  when  the  speed  is  constant,  the  rate  of  change  of  speed 

is  «ero;  that  ia,  when          7  -  72  .      JH_  .  0  t  (56) 

dt 


.*> 


.o 


..  > 


' 


Than,  from  equation  (55), 


Dividing  this  Toy 


f.5-7; 


In  order  to  determino  tho  raltw  of  V 

let  v;  =  tv 

»  »lk 


Let 


_ 

3 


+  r  » 


VY 

is,  let 


is 


How 


or 


-  26  - 


t 


(€,1) 
(62) 

(63) 


(68) 


o  «= 


*  l^x  *  tj*  ** 


K 


V  -«< 


I 


- 


- 


ItJ 


•»  Vf  , 


In  general,  '»;  will  have  throe  distinct  values,  et  least  one 

of  which  Is  real*  That  la, 

WV  q.W  +  r  =  O  [63) 

has  tJureo  roots,  and  at  least  one  root  is  real* 

Let  m  be  a  real  value  of  • 


and  lot  n  be  a  real  value  of 


n  *  n  will  be  a  real  root  of  1. 


Let 

The^the  roots  of 
are 

And,  ainoo 


ths  values  of  Tg  In 


are 


(a)  -  --3- 


-A  (a>n, 


vz  = 


xj,  = 

/ 


--** 


It  10  Men,  from  equation  (73)  and  tho  roots  (77),  that, 


« 


and  the  values  of  V7  arc   2?n,  -^^  -m  ,   all  real. 

Ihat  is,  all  the  values  of  W,  and  conae'iuontly  of  72  wo^a  1)e  reftl 


(76) 

(63) 
(77) 


(75) 


'-- 


-  27  - 


"ic  owl**?  Jtan  A  erf  a  fol  &n§ 


•  V  •«*  *  wf  IlJbv    K  '  at    /not 

=--  =  u> 
C>  >s  -\  -v  Hi  ^  -v-  "^VM  ^°  »fooi  tf9j9lB 

9tttfwtt         fcffd 
. 


(«,) 


* 


^  JL-   =  ,cx 

c* 

-V       -  -     m    p 


.    j<;  «.  -v  $«*«>  -V  JE  -- 


ifelch  migjvfc  loot!  to  umbic'tiity.    Howevor,  in  the  problom  at  hand,  an 
investigation  of  the  r-'mgoa  of  values  of 


show*  that.  In  ordinary  oases. 


In  other  words,  the  solution  yields  only  one  real  value  of  7g  sad  two 
imaginary  values*  The  real,  or  principal,  value  of  Vg  is  the  actual 
balancing  speed* 

2  -  Acceleration 

Tho  next  step  is  to  derive  a  formula  by  which  the  speed-time 
relations  can  be  detorminefl  for  periods  when  the  speed  of  tho  train  is 
increasing  or  decreasing  with  rated  terminal  voltage  applied  to  the 
motors* 

In  the  preceding  section,  the  fundamental  aquation  (1)  was 
adapted  to  the  condition  for  full  voltage  applied  to  the  notor  terminals, 
sod  reduced  to  simplest  terms,  namely  t 

Then  it  was  shown  that,  when 

either  \4  =  />/  =  -^-+7n-fn.  ,  (ya) 

t  i  *f  «  /          \ 

V2  =  jOj..  =.—-**+  win  +  &  n  ,  (73} 

*  m.  /      *•  \  '         / 

i  -  /°3  -     '—  +  <*>7n  +  t-m  (30) 

were  substituted  for  T  in 


then  Vt3+  liVS+tfiVscti^  O  .  (sa) 


-  28  - 


• 

' 


' 


. 


al 


ftUMI 


t«s-f 


• 


-v 


-v  5_  —  = 


-v 

w 


Therefore 


Alao  a2  =  -/0,/ct,^   ,  (36) 

,  (ar) 

(as) 


Introducing  tho  relation  (85)  into  equation  (55)  renders 


That  is,  f(V>  =  — 

g(v)   (i/ 

aay  be  resolved  into  partial  fractions,  thus 


vbaro 


_ 

' 


from  the  equations  (90),  (94)  and  (95),  it  U  aoen  that 


Integration  then  reader  a 


where  C2  la  the  oonatant  of  integr&tion* 


-  29  - 


transposed  is  _  (V-hJdr  _  ,     } 

>il"  ^-/3-)^-/°^^-/»J 

Let    /ri/;  =  v-h*  (91) 

and       y  (V)  = 


(93) 


l»S)  < 

V"^  ,    ^C-N, 

(86) 


(ae) 


(ea) 

(<»)  -     fZ^  "^^ 

M 

(ee)  .      .,e^N,scs  VNe  * 


&. 


|»^ 


JS.       !*15  1, 

M 


-^^ 


t  =   l**fc-/* 


(/ot) 


It  (7«V  ,  t-tj)  IB  a  point  on  the  curve;  that  is,  if 

t  **t,  , 


the  time  t  at  tfiich  the  speed  will  attain  some  other  value  7  is  given 

by  tflie  equation, 

t-t.  =  - 


In  oonnon  lo^arithna  and  expanded  for  convenience  in  computation, 

t  =  t,  +  •^^\l*lofr(V-/°,)  +  r 

-  h  log,,  (y,-/°'  - 
eqwtions  (101)  and  (103}  and  letting 


,=  --  =  4  — 


,  from  equations  (105)  and  (108), 


t    =    C*  + 


JogsrCv-p,)  +  7njog,.(l/-/>f)  + 


(109} 


p  -  ooiariiiG 

stations,  descending  grades,  s lowing  down  for 
crossings,  etc*,  it  is  generally  advantageous  from  the  standpoint  of 
energy  econoray  to  shut  off  the  eloctric  power  supply  and  "llow  the  train 
to  coast  for  a  period  before  the  brakas  are  applied*  Under  this  con- 
dition, the  motors  4o  not  supply  any  power  to  tho  train;  in  fact,  they 


—  30  ~ 


.  1}  ^,  ^_Ni) 


^ 


«c 


** 


--l  *• 


( 


- 
A 


s      - 


r^.-^V 


draw  a  oertr.ln  aran.ll  amount  from  the  kinetic  energy  of  the  train  in 
ordor  to  ovoroosie  tho  friction  and  windage  losses  in  the  motors  and 
gear*.  However,  this  draught  is  usually  negligible,  *o  it  my  be 
assumed  that  the  tractive  effort  input  to  the  train  is  zero  during 
coasting.  Alto,  by  tho  abovo  definition,  the  br.:kinn  effort  is  zoro 
during  coasting.  Hence,  with 

1-0  (110) 

and  B  -  0  ,  (111) 

the  fundamental  equation  (1)  reduces  to 

=  0.01  [-C-G-^-  0.03  V  -  ^^  (,+  ^L)  V*}        (,12) 


55io  Talue  of  tt  will  bo  positive  or  negative  according  as  the 
train  is  going  up-hill  or  clovm-hill*    fhe  magnitude  and  algebraic  sign 

of  the  vnltuj  of  5  will  determine  ^Aether  the  acceleration,  £3L  ,  is 

dt 

positive,  negative  or  zero*  pressed  algebraically  t 
0   if   G  > 


if     G   =- 

if       G     <- 


In  •quation  (112)  lot 

ct3  =    0.01  (C  +  G  +  ^=r)  *  (lie) 

/&,  =  0.0003  ,  (in) 


-dt 


V*)    ,  (119) 


-  SI  - 


• 


-  <   a 


o 


-« 

-o 


- 


A* 


eat  ttw  law** 


Honco 


('**) 


('**) 


There  aro  throo  possible  solutions  of  equation  (125) •  Ihe 
algebraic  sign  of  ^—J^Jdet  ermines  which  Is  the  proper  solution. 
The  vnlttes  of  /33  and  i9  are  positive  r.s  long  aa  the  train  is  moving* 
Hotv3T9r,  the  value  of  or,  is  positive  or  negative  according  as  G >  -(t 


or 


lTovortholosf;,ct,  ,/33  and  ?3  are  Constanta,  fixed  by 


track  and  operating  conditions  so  it  can  readily  be  determined  Whether 


et, 


Oase  I 


If 


the  solution  of 


ia 


sin 


,'  ,    (>27) 


or 


V+ 


OK  '»( 


_ 

J 
* 


-2     .  -/I"   ^7r3v-i-/^*   1  _  c^    /  .  _\ 

v<^«3/3-/«/  S/A'  ^/,r^^*^>»*^«*jj  "  *»  '  ( 

~ 


- 


, 


. 


( 1£v  -    4» 

V&    tn* 


. 


•   » 


t  O    <  -* 


' 


~ 


l«x)   4- 


e\-       SX: 


FV 


v 


where  time  la  expressed  in  seconds  and  the  Inverse  sine  (sin  )  in 
radian*  • 

Since,  in  trigonometric  tables,  sines  aro  conr.iled  as 
fc&iotiona  of  angles  expressed  in  degrees,  it  is  more  convenient  to 
modify  the  formula  (130)  ao  that  the  inverse  sine  con  be  derived 
directly  from  the  tables  in  degrees  and  deoimal  fractlona  of  a 
degree*  This  ia  accomplished  by  introducing  the  factor,  57*296, 
tho  number  of  degrees  in  a  radian.  Equation  (130)  thonbecc 


t  =  —    wr  *****    c; 


-I        -:-n' f   *f*V+*»     1  ..  Ci     (/32) 

.6V*<*3t3-tt*'  \y+/3(w+0,v+<*,)]  13   ^  ' 


z», 

where  time  ia  expressed  in  seconds  and  the  inverse  sine  in  degrees* 

If  (t-t  ,  W3)  be  a  point  on  the  speed-time  cwve  during 
the  coasting  period,  that  is,  if 

V  -  7a  when  t  -  «3  |  (183) 

tho  time  t,  at  which  the  train  will  attain  any  other  speed  T  by 
coasting,  is  given  by 

t-t,= 


.-/        3*       ,  -   , 

"   ?  \/**£r+*v+«   '  ^ 
Combining:  •qn&tions  (132)  and  (135),  and  letting 

s 

c3  =  -^  , 


' 


-* 


I 


9*1 


BPOCB  Ml    ellf 


JV  N»  f  UWMJ        *c»  -vj 

vM  1\_  •  M  y«* 

^  j^ja»      * 

^Mrft^itakf   9^ttr  --      *      • 

-  /     £W 


r 


•M 


If 

(125)  becomes 


The  soltxtion  of  this  Is  ,  / 

1*t    = 


-f2--^-    =    O       ,  (139) 


(141) 


or 

If  V  -  V3      tghen      t  -  tp     ,  (133) 

tho  tine  t,  at  which  tho  train  will  roach  auy  other  speed  T  by  coasting, 
is  given  by  a 


or 


Combioing  eqtjy.tion8  (142)  and  (144),  and  letting 


Than 


t    = 


f/»7) 


III 


If 


-^--^L     < 

/3    ^/a1  r 


(>+*) 


i«here  tho  parallel  vortiwal  lines  signify  that  the  absolute,  or 
niariaal,  value  or  the  quantity,  that  they  enclose.  Is  t&Ken  with 


-  34  - 


. 


*  £  = 


. 

~e\ 


. 


the  positive  algebraic  sign* 

Substituting  tho  relation  (149)  into' equation  (125) f  gives 

-Ut  = 


The  solution  of  tills  Is 


lot 


9 v 


*,f3-&\ 
In  cornaon  loganlthaa,  in  order  to  facilitate  confutation. 


{,    •— 


2. 


If 


t     -    tg 


(133) 


the  tisio  t  ,  ftt  vthlcii  the  vraln  will  attain  any  other  speed  T  by 
coasting,  is  given  by 


•f         f 

3  " 


so  that 

t    =    t 


" 


equations  (154.)  and  (156) ,  and  letting 

c 


C3  =  4  - 


t  =  c3  + 


flf0\ 

MI 


2. 


-  35  - 


• 


•.'•  arf 


- ^t»» IV 


. 


. 


oe.s 


r  . 


. 
~. 


. 

. 


RV 

-  «£v-  te^-  £<»£iV  j 

rJt&'-.Y.t^vi 


.. 


oe.s 


When  tho  brakes  are  applied  to  a  train,  the  power  surply  la 
uawlly  8t>-.iJG  off  00  in  coasting,  «o  *k®  Input  tractive  effort,  P  In 
equation  (1)  la  zero  during  the  braking  period,  nonce,  aquation  (1) 


-  0.03V-         0+V*.    060) 


Bratoea  are  most  comnonly  applied  for  tho  purpose  of  retard- 
ing the  motion  of  a  train.  However,  occaeionplly,  as  during  tho  descent 
of  grrdes,  the  braloas  are  partially  applied  and  the  speed  allowed  to 
increase  thoucft  not  to  such  an  extent  as  It  would  if  tho  brakss  wore 
not  applied,  dtoeroforo,  in  order  to  be  general,  the  formulae  for  spt«A- 
tine  relations  during  braking  •not  provide  for  positive,  zero  and  naga- 
tive  acceleration.  &qpresaed  algebraically t 

$<o    *    G>-[B  +  C^+0^+^0+^'. 
$-0    if     G=-[B  +  C+^+O.OSV+!^(,+  ^ 


>0     if       G<-  fo  +  C+  ^+  0.03V+  ±°°?±(I+>±±)V*}  -          M 


dt 

She  acceleration  is 
A    -  ^p  =    O.O/\-B-C-G-^~  0.03V- 

ot  d4=    O.OI(B  +  Q.+ 


#,  =    0.0003      , 

0. 00002 X  fl       N-l^ 
'U   =  -7=-     -(^—jQ-f      ' 

dV  . 


-  36  - 


-j- &-*-*-- 


- 


Kt{l-VV  ^.^XSOO.O        .,___ 

"  L    ^~oT      ^~  • -Meo.o  - 


•-  -  f- 


•v  ft  1  \O.O      =  JD 


,    eooo.o  =  £>s 
M  r    C^-^M5^  m  + 

(«»\)  -     ^M^-N^^-^y- 


A  comparison  of  equations  (116)  to  (119)  and  (165)  to  (16S) 
Shows  that  fcho  spood-ti'ne  relations  during  the  coasting  period  are 
different  from  those  in  the  bracing  period  only  in  tihat  B  f  0  in 
the  latter;  that  la 


=  o.oi 


while 


a.,  =  O.Oi  '(C  '  +  G  + 


Hence,  ainco   and   are  both  constants,  the  solutions  for  the  two 
periods  v/lll  be  similar  in  form  and  tho  reuniting  formulae  for  the 
braking  period  onn  be  written  directly* 

speed-time  curve 


Thus,  if  (t-t  ,  V«7  )  ^°  a  PoJjl* 


in  the  brruring  period,  that  is,  if 


V  -  7. 


t  »  t 


.  . 

4  4 

the  ti:«»  t,  at  -nhioh  tho  speed  xrill  attain  somo  other  value  7  under 
constant  -"--.ilication  of  tho  brakes,  is  given  by  tho  following 
forrrnlaas 


(169) 


|f 


-     >  0   , 


;,;' 


Q  = 


In  these  formulae,  tirao  is  expressed  in  seconds,  speed  in  miles  per 
hour,  and  Inverse  sines  in  degrees* 


-  37  - 


*  t 


-vo 


r 


Iflsia  60  lllw  sLc lie» 


v. 


-gi£|  Jni6 


^ 


• 


e-, 


Case  II 


If 


—     O 


t  =  t  + 


Letting 


("*) 
Q77) 


Case  III 


If 


<o    , 


so  that 


2.30 


, 


-*  , 

--     -          ~       " 


Letting 


c.  =  4  - 


2.30 


(ire) 


0?o) 


?.3Q  « 

^rrm*0*" 


+  /Q*.+2l4.V\ 


Special  Case,    V  "  0 

In  stopping  a  train  by  the  application  of  friction  brakes, 
if  the  speed  is  VA  at  the  instant  t^  in  the  braking  period,  the  tiir» 
t,  at  which  the  train  will  atop,  is  given  by  letting    V  "  0    In 
equation  (171)*    Then. 

/ 


za. 


sin 


.  -/  r  &4 

-  sin  \-j==~ 

[V 4K+1, 


(,33) 


^^rNJS* 


.A^Nt^S 


, 


P®v) 


\- 


-V*s 


^* 

5i^M 


']* 


O&.S 


- 


1C  *  *  -  J» 


- 


-  ^  =  J> 


(68V, 


p  -  goaauiT   o?  PRINCIPAL 


In  conclusion,  the  principal  formulae  are  grouped  together 
in  order  to  facilitate  reference  to  them* 

The  acceleration  formula  f  or  fundamental  differential 

•qoatlon  of  train  speed  is 

ff>          t 
-  0.03V-  - 


'^r)y'\ .   (f) 


9w  formla  for  the  tractive  effort  input  to  the  train, 
rated  voltage  la  oppUed  to  the  motor  *;0rainalst  la 


For  the  starting  period,  that  is,  with  constant  tractive 
effort  innut  to  tho  train, 

**- 1 «  2-3^ 


-A-zt V) 


(39) 


If  V0  -  0  and  t0  "  0 


4  = 


During  acceleration  with  rated  voltage  applied  to  the  motors, 

,        /  T     y-fi 

*  ~  •«  +  ~J~  «« *°^e  TTTT"  +  " 


(47) 


T, 


During  conating. 


t-  t,+  j— 

*   za.B-V^K^-ft 


proridefl 


-  39  - 


. 


CfeOQ*  J0 


-  N  &<XQ  - 


\0.0 


^  ( 


^ 


.      0  <  _ 

'  * 


'   «ft  to--  -ni 


* 


.     .  i.  •  • . 


IX 


*-4 


if 


^L-^L  <0 

i*   *&  N 


Durlnc 


=  t  + 


provided 


sin 


—  sm 


If 


t  =  t  +  - 


(/S6) 
0+8) 


(170) 

0*4 


(/SO) 


if 

If  the  speed  steadily  deoreaaoa  to  ?  =  0  taider  the  application  of  the 
the  train  will  atop 

v-     / 

^«.6  V^^^rTOT 


-40  - 


**> 


v 


X 

*^ 


x^ 


Wi) 


r  *  >  - 


o  = 


^NJ5 


O  x    ~ 


•  •;• 


mt  — 


m*-. 


IV 

EXAMPLES 

These  examples  are  given  for  the  purpose  of  illuatratine  the 
methods  of  applying  the  foregoing  formulae*  For  simplicity,  they  are 
all  computed  for  the  same  train. 
Train  Data 

Htnriber  of  cars,  H  "  6 
Motor  cars, 

Trailers ,  5 

Motors  per  motor  car,  2 

Number  of  motors,  II  •  6 

Weight  of  three  motor  cars,  3  Z  28  «  84  tons 

Weight  of  three  trailers,  3  X  22  »  66  tons 

Total  weight  of  train,  T  •  150  tons 

Gross  train  weight  per  motor  150  f  6  »  25  tons 

Area  of  projected  cross-section  X  •  110  sq*ft< 

Average  starting  current  per  motor  350  amperes* 

Motor  Characteristics  Fig*  2 


.• .  •    .       •/.-.. 


•\, 


-41  - 


i 

:  XW»   tT  "0*      • 

nt    <MBB«   crt     101    IT 

. 

tB*U3O    10    ' 
,  »1JU 

, 

,-iciS  v.ai-oir  • 

,«W*0«J  '  .-JU 

-  '      -     :  Offl  Wtri*   ^O    Jl^l 

••  38  X  5  <n£  •e^dt  1e  -  iv 

r.  .,  -jjio-,*  j 

;S  »  «  1  C3I  «^0£  •%!•«  : 

'    •  •  X  • 

• 

* 


. 

How  much  tl-ne  will  be  consumed  in  the  starting  period  on 
level  tangent  track,  that  la,  to  accelerate  the  train  from  standstill 
to  17  «2  miles  per  hour,  the  speed,  on  the  mot  or  a1  normal  speed  curve, 
corresponding  to  the  average  starting  current  of  350  anperoe  ? 

Prom  the  character  i  at  ic  curves,  Fig»  2,  the  tractive  effort, 
corresponding  to  350  amperos,  ia  5000  ll>s.  per  motor.    Hence,  the 
tractive  effort  per  ton  of  cross  train  weight  is 

P  -  5000  /  23-200  Ibs.  po-  ton. 

Jinoe  the  track  la  lovel  and  straight,  and  the  tandOM  are  not  applied 
during  starting,  B-C-G-0       • 

in  equations   (24),   (25)  and   (26), 


a,  = 


=     1.96     , 

-3 

/5,   =    O.O003  O.3x/0      , 

O.OOOOZ 


/SO 


Then  formula  (47)  Gives 


-0.0003x17.2  +  /7.Z 


=  a.Q 


-42  - 


»-<A*  eo.o  ^- 


~"     V 


2  -  Balancing  Speed 

VShat  will  be  the  balancing  speed  on  a  one  per  cent  up-grade  ? 
On  the  straight  line  AA,  which  approximates  tho  motor  power 
output  current  curve,  Fig.  2, 

P*  -  160  ••hen  I  -  325 

and  P1  -  49.4    when    I  -  100. 

Substituting  these  values  in  equation  (2)  gives 

160  -  h^  +  325  h£ 

and  49.4  -  h|  +  100  h^  • 

Solving  these  as  simultaneous  equations, 

110.6  -  225  h'  , 

hj  -  0.4915 
and  h|  -  0.250   • 

On  the  straight  liao  BB,  which  approximates  the  tractive 
effort-current  curve,  Fig.  2, 

P1  •  4980  when  I  •  350 

and  P1  «  880  when  I  "  100  . 

Substituting  these  values  in  equation  (10)  gives 

4980  -  h,  +350  h. 
3        * 

and  880  «  hg  +  100  h^  • 

Then  4100  •  250  h    f 


and  h_  -  -  760    . 

Substituting  these  values  of  hj  ,  hg  ,  hg  and  h^  in  equations 
(20)  and  (21)  gives 

-  43  - 


' 

JL 

. 

< 

' 

« 

*%i 


:&r     068  -   ' 
*    _ll  *  Uce* 


r  if!      -     OI^^H 


ni  ±fi  IUBJ 

|OS) 


iff  *  *   .       760  X  0*4916 
*  °'25  * 


-  453.4 


-  18.0 
Ihua  aquation  (22)  glrot 


F  .   463.4     9 
T  -  15.0 


lor  oae  por  cent 

5  -  8000  aln( 

-  20.0  . 
(50),  (52),  (53)  and  (54)  give 


=      -ZZxIO6    , 
O.OI 


-      -  374.8  x/03    , 

A   =    - 


=     /0.74-x/O3     , 


=     -  7.364-     . 


(61)  ana  (62)  give 

A 

li     aft  \ 

a     *    10740  - 


10.74  X  103 


and  r    -    -374800 


I 
10740  X 


3 
-369.9  X  103  . 


HelPtlons  (74)  and  (75)  give 


-  44  - 


. 


t,  \ 
^•°-l- 


[369900  /  (369900)*   t    (/074-oJ* 

_^_ ^—    ^       i  __— ^— — —  -^-   i 
2V*  27 

=        74.  /  7      ,  

f36S9OO  / (369900)*        (/O74O)3 

11    =       —        V~~   '*  ~^T~ 


Honc«,  t>7  foRnoloe  (78),  tho  balr.no  ing 


_ 

' 


74J7-4-e.il 


=    29,^52.    niloa  per  hcrur* 


3  -  Aocelerati 

If  the  trj-.in  is  aooolcwtted  fjpom  the  oloao  of  the  atarting 
period  on  a  one  p«r  oent»  grade,  Tiow  ionpr  after  starting  from  atand- 
atill  will  the  speed  bocomo  28*0  nilea  per  hour  ? 

From  the  preceding  eaoanplee  and  equr.tiona  (76),  (78),  (79) 

and  (80), 

t,    =     8.8     , 

V,    =     /7.a    , 
V    =     28.0  , 


=      -  13.38  +J24.3O     -, 

-y 

=     —13.58  -j  24.30     . 


From  «<n»tiona  (96),  (97)  and  (98), 

—13.58  -t-jZ.4-.aO-  1  5.O 


3(-/3.5S+j  Z+.30)*  +  Z 

=      -  O.O033B+j'O.OOiaZ 
-  45  - 


c^v/  .MTT\ 


^     <x 
€/ 

'. 


-    --^  -        -s^-«  *         • 

l- 


. 

•  ftawi 

c 

€}  tarn 

«      8.8     ^^     vi 

,    S.\\ 

«   0.8S     =     \I 


_ 


3(ZB.SZ)*-I-  Z(-l.364Xza.SZ)  +  107+0 
O.OO/03       , 

-/3.  Sa  -JZ+.  30  -  /$.  O 

~\+t074O 


O.OO338  -jO.OOieZ      . 


thase  value*  la  aqmtlon  (104) 


=    8.8  + 


-22 


+(-o.  00336  -t-jo.ooiaz)  ?oge  y^ 

+(-O.OO33B-j 


(I-J.Z 

(za.oo 


t     =    8.8- 


!£. 

22 


0. 00103 


0.^5Zx 

11.32* 

+  (-O.0033&+jO.OO\&Z)logifi(4I.SQ-jZ4.30) 
—  (-O.OO33B  +/O.OO/SZ)  10ge (30.78  -J24.3O) 
+  (-0.00338  -JO. 00/82)  lofcfrl.SB  +J24.3O) 
-  (-O.O0338  -JO.OOI82)  loge(3O-7B  +J2.4.3O) 


POP  logwittane  of  the  connloa:  qur.ntltioa,  in 
1qge(x+jy)  =  Jog>\X+jy\  +jtan%  +  zjmTT  ,  m 


infinite 


liowov  r,  oinoe  m  lo  any  intocer,  thU  fowrala 

/        .  r^<J^ 

imifttr  of  v.luoo  of   lofrlX+jy)     .     nat  unly  tto«  pgia^tpul  trluos  of 

•^    "^^^ot_ 
Jtflnitt  li^mnli  ar«  oonoernetl  In  apeecS-tlMe  detormlwtiona,  so 

the  last  t«rmt  a  J  •  *  ,  wy  b«  tMglootedi  that  is  let    •  -  0  • 
Iha  rMOltir*?  nolublon  UMB  it 

t  •  8*8   *  230*7  »  239*5     aeoor^bi  9 
4*00 


-     .    - 


-oc 


( 


(oe .  vs\-  e%>\v)  «^>\  (ssvoo  .o'i 


\  4 


-  -v 


-  8.8    = 


Ttmys  -v 


V 

\s-v 

^ 


• 


&-, 


If  thu  pcwtir  is  shut  off  whan  the  speed  re:  dies  2C.O 
per  hour,  iaul  tho  train  is  allowed  to  coast  tip  ths  cao  per  o«nt  grate, 

long  i-ftci-  the  train  a  tar  toil  v/ill  tiie  apoed  bocoiae  10*0  railea 
per  hour  ? 

(11C),  (117)   and   (118)  ffivo 


f  S~O 

or,   =      0.01  ' ' 


O.Z403     , 

>0,    =•        O.OO03    , 

O.OOOOZ 


,50        "°"  70 


O.os  x/cT 


—     /O.9x/03  >O    . 
fflxerefore,  fomsula.  (135)  applies  and 


5//1 


_-6 

—I 


lSxl66(zZxl6"\  2&.o    ojxnt  '28.o-o.Z4V6, 
2  X  22  X  /O^X/o  +O-OO03 


-sin 

-\J+X 22.* A»*(22  X  If  -h  O.OOO3X  fO.O +  O.2.4-O8) 

=  233.5  +70.6  . 
=  3/O  5econ</s  9 
=  *5.I7  minutes  » 


• 

. 


e* 


,  eooo.o      = 

SOOrK>JD 
-\ 

sa      = 


o\xs&x-s 

."^?V 

GoOO-O^- 


;  »'.    -t.  I 


~^7d^  *  t-ees 

me- 


Prom  the  time  the  speed  reuohoa  iO«G  miles  por  hour  until 
the  ti'.'ln  stopa,  an  average  broking  effort  of  200  potmts  per  ton  of 
&roas  train  w*igab  la  applied  by  friction  brakes,  the  grade  continuing 
at     +  1*00  per  cent,    now  nruoh  time  will  hare  been  oozunmd  In  making 
the  run  froa  start  to  stop  ? 

?rom  oeuationa  (165),  (166)  tuoal  (167), 


=    z.Z4oa  , 

=      O.OOO3    , 

o.ooooa 
/so 

-     22x70"* 
2.Z40Q 


110(1+ 


0.09  x/0* 


ZZxlti* 
=       lOZxIO3  >  O 


Therefore  fornjula  (1R3)  applies 


t     =    3/0  + 


SLH 


-sin 


=     3/O+  4.8 
=      31-5 


.o +0.0003 


O.OOO3 


+O.OS  -v 


\ 


,  eooo.o 


—        v> 


,^s^v  ^ 


,  O 


•    : 


, I 


-    -V    0\( 


*o\  ^&. 


s.v 
.«*•-.-.•          'i."^ 


14  DAY  USE 

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'  O 


General  Library 
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